Question: $ -1.\overline{97} \div 0.\overline{46} = {?} $
Explanation: First convert the repeating decimals to fractions. $\begin{align*} 100x &= -197.9798...\\ x &= -1.9798...\end{align*} $ $\begin{align*} 99x &= -196 \\ x &= -\dfrac{196}{99}\end{align*} $ $\begin{align*} 100y &= 46.4646...\\ y &= 0.4646...\end{align*} $ $\begin{align*} 99y &= 46 \\ y &= \dfrac{46}{99}\end{align*} $ So, the problem becomes: $ -\dfrac{196}{99} \div \dfrac{46}{99} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ -\dfrac{196}{99} \times \dfrac{99}{46} = {?} $ $ \phantom{-\dfrac{196}{99} \times \dfrac{46}{99}} = \dfrac{-196 \times 99}{99 \times 46} $ $ \phantom{-\dfrac{196}{99} \times \dfrac{46}{99}} = \dfrac{-196 \times \cancel{99}} {\cancel{99} \times 46} $ $ \phantom{-\dfrac{196}{99} \times \dfrac{46}{99}} = -\dfrac{196}{46} $ Simplify: ${= -\dfrac{98}{23}}$